1. Introduction

Since the first attosecond pulse was obtained experimentally, various attosecond spectroscopies have been reported and implemented to probe ultrafast electron dynamics in matters [1–4]. As one of the outstanding representatives, attosecond transient absorption spectroscopy (ATAS) provides an unprecedented temporal resolution for real-time observation of electron dynamics on its natural time scale [5, 6]. ATAS usually arises from the interaction between an attosecond extreme-ultraviolet (XUV) pulse and a matter target dressed by a few-cycle infrared (IR) pulse, and records the spectral change of the XUV pulse passing through the target versus the delay between these two pulses, which reveals the transient dynamics in the interaction.

The first successful application of the ATAS is the measurement of the valence electron motion in krypton ions generated by the controlled few-cycle IR pulse [6], which motivates a great deal of experimental investigations based on the ATAS technique to achieve the ultrafast measurement from the gas phase to the condensed matter. It has been demonstrated that the ATAS is successfully used in numerous areas, such as revealing the interference of transiently bound electron wave packets [7], probing the optical Stark shift of the laser-dressed 1s3p state of helium atom [8], reconstructing the electronic and nuclear dynamics in hydrogen molecules [9], resolving the ultrafast band-gap dynamics in silicon in real time [10], and others [11–16]. Besides these advances in experiment, theoretical calculation of the ATAS was realized by a time-frequency method [17]. Using this method, the nuclear-motion effects of molecules in the ATAS were analyzed [18] and the quantum beats were first observed in the simulated absorption spectrum of helium [19]. Theoretical development enables the further investigation of the rich dynamics of ATAS [20–25].

To date, theoretical researches on ATAS are all focused on the use of a chirp-free XUV pulse to pump (probe) an excited electron wave packet which is probed (pumped) by a few-cycle IR laser [18, 21, 24], and the chirp characteristic of attosecond pulse has been ignored. Nevertheless, chirp is difficult to eliminate in attosecond pulse generated by high harmonics due to the intrinsically laser-intensity-dependent dipole [26]. Moreover, recent studies have demonstrated that chirped laser pulses possess selectivity of controlling the population transfer in the process of two-photon double ionization of atoms [27, 28]. Taken this into consideration, we guess that chirped attosecond pulses may have influence on ATAS which contains a similar two-photon process as pump-probe absorption measurements. What the additional effect remains to be addressed.

In this work, we report the first study of attosecond pulse chirp effects on transient absorption spectrum. Firstly, we introduce the chirped Gaussian attosecond pulses with different linear chirp rates instead of chirp-free pulse to calculate the ATAS of laser-dressed helium atom by numerically solving the fully three-dimensional ti spectra exhibit almost the same structure for different chirps, obvious relative translation between them along the delay axis is observed, which gives evidence of the existence of chirp effect on ATAS. Secondly, considering that the 3D-TDSE calculation is time consuming and difficult to reveal inherent physical mechanism, we further employ the three-level model which is a more efficient method to analyze the contribution of specific electronic states [19, 29]. According to this model, we have successfully reproduced the chirp effect and through the analysis of the IR-field-dressed eigenstates, we find that chirp effect is associated with indirect two-photon absorption pathways involving the absorption of an XUV photon and then the absorption or emission of an IR photon via a set of the virtual states, so that the electron can transfer from the ground state to these eigenstates.

2. Theoretical model and numerical method

The ATAS of helium atom in the presence of the dressed IR pulse is calculated by numerically solving the fully 3D-TDSE under the single active electron approximation. This treatment has been widely used in many previous works and proved effective in reproducing main features of ATAS in helium [14, 17, 23]. In the Cartesian spherical coordinate, the fully 3D-TDSE is expressed as

The ATAS is always implemented upon scanning the relative delay of the IR pulse and the XUV pulse. We assume that the two pulses both have the polarization direction along the $z$ axis. In the length gauge and dipole approximation, the $V_{\text{I}}(\overrightarrow{r},t)$ can be described as $V_{\text{I}}(\overrightarrow{r},t)=zE(t)$, where the total electric field is related with the corresponding vector potential $A(t)$ as $E(t)=-\frac{d}{dt}A(t)$. Here the total vector potential of two pulses can be written as $A(t)=A_{\text{X}}(t-t_{\text{D}})+A_{\text{IR}}(t)$ with the delay $t_{\text{D}}$. The time-dependent vector potentials of a chirped XUV Gaussian pulse and a chirp-free IR Gaussian pulse can be written respectively as [31]

A detailed description of the numerical method used to solve Eq. (1) is reported in [32]. In brief, the time-dependent wave function $\Psi (\overrightarrow{r},t)$ is firstly expanded into a series of spherical harmonics. As a result, Eq. (1) can reduce to a set of coupled equations between the different angular quantum numbers for the radial wave function, which is further discretized by the finite-element discrete variable representation (FE-DVR) method with the advantage of providing block-diagonal sparse matrix representation of kinetic operator and the diagonal matrix representation of the effective coulomb potential. The temporal evolution of the wave function is carried out by the Arnoldi-Lanczos algorithm.

After the fully 3D-TDSE is solved, the transient absorption spectrum can be achieved by calculating the single-atom frequency-dependent response function [17]

3. Results and discussion

In order to reveal the chirp effect of attosecond pulse on the transient absorption of IR-dressed helium atom, we have firstly calculated the single-atom response function $\tilde{S}(\omega ,t_{\text{D}})$ using the fully 3D-TDSE model upon scanning the relative delay between the IR pulse and the XUV attosecond pulse.

Figure 1 shows the calculated ATASs in helium for three different chirp rates of the XUV pulses: (a) $\xi =-4$, (b) $\xi =0$ and (c) $\xi =4$. Note that the negative (positive) delay represents the XUV pulses arrive before (after) the IR pulse. In the calculation, the XUV pulses have a central photon energy of 24 eV, a transform-limited duration of 300 as and a peak intensity of $1\times {10}^{12}{\text{W/cm}}^{\text{2}}$. These pulses have a bandwidth of 6 eV, which can promote the electron from the ground state to all singly excited states below the single ionization threshold ($I_p$) and to low-energy continuum states of helium atom [23]. The IR pulse with the 800 nm wavelength has the duration of 10 fs and a relatively moderate peak intensity of $1\times {10}^{13}{\text{W/cm}}^{\text{2}}$. We point out that a relatively moderate IR intensity of $1\times {10}^{13}{\text{W/cm}}^{\text{2}}$ is a proper to make the chirp of attosecond pulse readily observable in absorption spectra. The weaker (~$1\times {10}^{12}{\text{W/cm}}^{\text{2}}$) or the stronger (~$1\times {10}^{14}{\text{W/cm}}^{\text{2}}$) IR intensity will induce too low contrast of absorption signals or the severe shift and splitting of energy levels, which prevents the possible observation in the experiment.

 

Fig. 1 Calculated single-atom response function $\tilde{S}(\omega ,t_{\text{D}})$ using the fully 3D-TDSE model upon scanning the relative delay between the IR pulse and the attosecond pulse with three different chirp rates: (a) $\xi =-4$, (b) $\xi =0$ and (c) $\xi =4$.

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One can see several similar features regardless of the attosecond chirp effect in Figs. 1(a)-1(c) as follow: (i) For large positive delays, there are the narrow absorption lines induced only by the XUV pulse, representing the excitation of the electron from the $1s$ ground state to all the $np$ excited states by single-photon transition. (ii) For large negative delays, a modulation of absorption following hyperbolic sidebands is referred to as perturbed free-induction decay [33]. (iii) For the temporal overlap region of the two pulses, the modulation of absorption with the period of half-IR-cycle and the light-induced structure (LIS) are present between the $2p$ and $3p$ resonant lines, which is firstly observed experimentally in 2012 [12].

Although Figs. 1(a)-1(c) exhibit almost the same structure, we can clearly reveal the attosecond chirp effect by extracting the absorption lines from Fig. 1 at three representative states: $2p$ (21.05 eV), $4p$ (23.73 eV) and a low-energy continuum state (25.55 eV). Note that the ground state energy is already set to zero. Figures 2(a)-2(c) respectively show the delay-dependent absorption signals near 21.05 eV, 23.73 eV and 25.55 eV, calculated as the integral of the response function around each state with a spectral width of 0.5 eV, for three different attosecond chirp rates. The primary feature included in the Figs. 2(a)-2(c) is the temporal shift of absorption curves with varying the chirp rate of the attosecond pulse, together with the shift distance dependent on the absorption line energy of the observation.

 

Fig. 2 Delay-dependent absorption signals near (a) $2p$ (21.05 eV), (b) $4p$ (23.73 eV) and (c) the low-energy continuum state (25.55 eV), respectively for three different attosecond chirp rates: $\xi =-4$ (red-solid line), $\xi =0$ (blue-dashed line) and $\xi =4$ (black-dot line).

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The fully 3D-TDSE calculation is time consuming and difficult to identify the underling mechanism due to the complete consideration of all electronic states and interaction dynamics. We further employ the N-level model which is an efficient and compact method [19, 29], to analyze the contribution of specific electronic states to the attosecond chirp effect on the ATAS.

As an example, we mainly focus on the absorption signal near the $2p$ state by choosing three energy levels composed of the $1s$, $2p$ and $3s$ states. In general, both the IR and XUV pulses can lead to the transition among these three states. In order to separate the contribution of the XUV and IR pulses, we artificially switch off the transition between $1s$ and $2p$ induced by the IR pulse, and the transition between $2p$ and $3s$ induced by the XUV pulse, which means that the XUV pulse only excites $1s$ to $2p$ and the IR pulse only excites $2p$ to $3s$. In this situation, we calculate the attosecond chirp effect by analyzing the absorption signals near the $2p$ state as a function of the delay under three different attosecond chirp rates: $\xi =-4$ (red-solid), $\xi =0$ (blue-dashed) and $\xi =4$ (black-dot), as shown in Fig. 3. One can see that the 2p absorption signals in Fig. 3 maintain significant features compared with the fully 3D-TDSE result shown in Fig. 2(a), including the same modulation period of 1.33 fs and nearly the same phase shift of peak positons with respect to the chirp rate $\xi$. This comparison demonstrates the three-level model instead of the full 3D-TDSE calculation is sufficient and valid to capture the important characteristics of the chirp-dependent ATAS.

 

Fig. 3 Absorption signals near the $2p$ state (21.05 eV) in the case of three different chirp rates: $\xi =-4$ (red-solid line), $\xi =0$ (blue-dashed line) and $\xi =4$ (black-dot line), calculated by three-level model. Other simulation parameters are the same as Fig. 1.

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For quantitatively identifying the chirp effect, we can define the delay offset $\Delta \left(\xi \right)$ for a given chirp rate $\xi$ relative to the chirp-free case as $\Delta (\xi )=t_{\text{D}}(\xi )-t_{\text{D}}(0)$, where $t_{\text{D}}(\xi )$ corresponds to certain absorption peaks selected by the arrows in Fig. 3. We use the value of delay offset $\Delta (4)$ to estimate the influences of the IR duration and the additional state couplings. We find that for both 5-fs and 10-fs IR pulses, the additional couplings beyond the 3-level model will result in the variation of $\Delta (4)$ less than 0.04 fs. It is expected that the contribution from the other Rydberg states such as $2s$ and $3d$ coupling to $2p$ state will become weaker with increasing the IR duration. As a result, the chirp-dependent absorption spectrum is mainly determined by the 3-level model for long IR pulse. If this effect is used to determine the chirp of attosecond pulse, the accuracy of the retrieved chirp can been estimated by $\delta \xi =\frac{{\Delta }_5({\xi }_0)-{\Delta }_3({\xi }_0)}{\frac{d{\Delta }_3(\xi )}{d\xi }|\xi ={\xi }_0}$, where ${\Delta }_5({\xi }_0)$ (${\Delta }_3({\xi }_0)$) is the delay offset for a given chirp rate of ${\xi }_0$ calculated by 5-level model (3-level model). Here the 5-level model consists of 1s, 2p, 3s, 2s, and 3d states. Since $d{\Delta }_3(\xi )\text{/}d\xi$ is approximately equal to 0.066 fs, $\delta \xi$ is less than $0.04/0.066=0.6061$.

Since the wavelength of the IR pulse is an important parameter in the strong-field process, we have further investigated how the IR wavelength affects the $\Delta (\xi )$ . Figure 4 shows $\Delta \left(\xi \right)$ as a function of the IR wavelength at five different chirp rates: $\xi =-4$ (red-pluses), $\xi =-2$ (green-stars), $\xi =0$ (blue-circles), $\xi =2$ (cyan-crosses) and $\xi =4$ (black-squares). It is found that $\Delta \left(\xi \right)$ is linearly dependent on the chirp rate $\xi$ and also changes with the IR wavelength. For the positive (negative) attosecond chirp rate $\xi$, $\Delta \left(\xi \right)$ increases (decreases) with increasing IR wavelength until reaching the saturation value in the long wavelength limit. Moreover, $\Delta \left(\xi \right)$ can change its sign at 420 nm position for all chirp values.

 

Fig. 4 The IR wavelength-dependent offset $\Delta (\xi )=t_{\text{D}}(\xi )-t_{\text{D}}(0)$ in the case of five different chirp rates: $\xi =-4$ (red-pluses), $\xi =-2$ (green-stars), $\xi =0$ (blue-circles), $\xi =2$ (cyan-crosses) and $\xi =4$ (black-squares) presents the fully 3D-TDSE results. Other parameters are consistent with Fig. 3.

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For a deep insight into the chirp effect on ATAS, we propose an indirect two-photon process based on the three-level model to interpret the above simulation results. The intense IR pulse can lead to the energy shift and the state mixing of the field-free energy eigenstates. To simplify the analysis and highlight the physical picture, we only consider the field-free eigenstates $2p$ and $3s$ affected by the IR pulse, which will result in the generation of two IR-dressed instantaneous eigenstates, denoted as $S_{-}$ and $S_{+}$ . Their energy eigenvalues can be expressed respectively as

Both $S_{-}$ and $S_{+}$ states are the mixture of the $2p$ and $3s$ states, in which the proportion of $3s$ state in $S_{-}$ and $S_{+}$ are given respectively by

The schematic of the indirect two-photon process is shown in Fig. 5, where the transition from the ground state $1s$ ($E_{1s}=0\text{eV}$) to $S_{-}$ eigenstate can occur through an alternative indirect $\omega -{\omega }_{\text{IR}}$ pathway associated with the virtual state $S_{-}^{+}$ with the energy $\Delta E+{\omega }_{\text{IR}}$, where $\Delta E=E_{-}-E_{1s}\approx E_{2p}-E_{1s}$. Because the mixed $S_{-}$ eigenstate include the $3s$ state component, this two-photon process is permitted without violating the dipole transition selection rule. The transition process for the electron excited from the ground state to the $S_{-}$ state requires that the helium atom gains the net energy $\Delta E\approx 21.05\text{eV}$. This can be achieved in the indirect way by simultaneously absorbing one XUV photon and radiating one IR photon. Consequently, the response function $\tilde{S}(\Delta E,t_{\text{D}})$ describing the absorption line near the $2p$ ($E_{2p}=21.05\text{eV}$) state can become the maximum value only when the XUV photon with the energy $\Delta E+{\omega }_{\text{IR}}$ reaches the maximal intensity. The chirp in the attosecond pulse leads to the time-dependent instantaneous frequency, so that the moment $t_m$ at which the $\Delta E+{\omega }_{\text{IR}}$ photon component reaches the maximal intensity is different for different chirp rates.

 

Fig. 5 Schematic of the three-level model including the $\text{1}s$, $2p$ and $3s$ states. $S_{-}$ and $S_{+}$ are eigenstates for the $2p$ and $3s$ states dressed by the intense IR pulse. The indirect two-photon pathway from the ground state $\text{1}s$ to the final state $S_{-}$ through the virtual state $S_{-}^{+}$ is also indicated.

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We can acquire the chirp-dependent moment $t_m(\xi )$ by performing time-frequency analysis to the XUV pulses for different chirp rates, as presented in Fig. 6(a). For a fixed energy $E$ and along the horizontal direction ($t$ axis) of Fig. 6(a), we can obtain a peak value which is related with the certain time $T$. This means that the $E$ photon component involved in the chirped attosecond pulse has the strongest intensity at the time $T$. We can plot $E$ versus $T$ for five different chirp rates, as shown in Fig. 6(b). The proposed two-photon absorption process implies that the efficient transition can occur when the XUV photon possesses the energy ${\omega }_{\text{XUV}}=\Delta E+{\omega }_{\text{IR}}$, whose position is denoted by the black-dashed horizontal line in Fig. 6(b). Consequently, the $t$-coordinate of the cross points of the horizontal line and those titled lines correspond to $t_m(\xi ,{\omega }_{\text{XUV}})$ $(\xi =-4,-2,0,2,4)$, as indicated in Fig. 6(b). It follows that the offset $\Delta \left(\xi ,{\omega }_{\text{XUV}}\right)$ for the $S_{-}$ absorption peak can be given by $\Delta \left(\xi ,{\omega }_{\text{XUV}}\right)=t_m(0,{\omega }_{\text{XUV}})-t_m(\xi ,{\omega }_{\text{XUV}})$.

 

Fig. 6 (a) Time-frequency spectrogram of the XUV pulses at different chirp rates: $\xi =-4,-2,0,2,4$ respectively. (b) Time-dependent photon energy which has the maximal intensity, extracted from panel (a) for five different chirp rates.

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According to ${\omega }_{\text{IR}}(\text{eV})\approx 1240/\lambda (\text{nm})$ ($\lambda$ is IR wavelength), we can further obtain the IR wavelength-dependent offset $\Delta$ for different chirp rates $\xi$, as shown in Fig. 7. The nice agreement between Fig. 7 and Fig. 4 supports the two-photon absorption picture. It is worth mentioning that there is a special IR wavelength 420nm for which the temporal offset $\Delta$ is equal to zero for all values of the chirp. We point out that the 420nm wavelength is related with the central photon energy of the XUV pulse. As already stated above, the required XUV photon energy is close to ${\omega }_{\text{XUV}}=21.05+{\omega }_{\text{IR}}$ eV and denoted by the dashed horizontal line in Fig. 6 (b). In order to make the delay offsets $\Delta$ is equal to zero for all values of the chirp $\xi$, the ${\omega }_{\text{XUV}}$ horizontal line through the dependence on the ${\omega }_{\text{IR}}$ should meet the S point shown in Fig. 6 (b). It follows that the special IR wavelength which leads to $\Delta (\xi )=0$ can be given by $\lambda (\text{nm})\approx 1240/({\omega }_S-21.05\text{eV})$. Here ${\omega }_S$ is the energy corresponding to S point, representing the intensity of the ${\omega }_S$ photon component will reach the maximum at the same moment for all chirp rates $\xi$. The ${\omega }_S$ is always equal to the central photon energy of the XUV pulse. In our simulation, the central photon energy of the XUV pulse is chosen as 24 eV. We can therefore obtain the special wavelength $\lambda (\text{nm})\approx 1240/(24-21.05\text{eV})=420\text{nm}$.

 

Fig. 7 The IR wavelength-dependent offset $\Delta (\xi )$ for five different chirp rates: $\xi =-4,-2,0,2,4$ calculated by an analytical model based on the two-photon absorption process. Other parameters are consistent with Fig. 4.

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The indirect two-photon absorption process can also be used to explain the chirp effect near the $4p$ state ($E_{4p}=23.73\text{eV}$) shown in Fig. 2(b) and the low-energy continuum state ($E_{\text{c}}=25.55\text{eV}$) shown in Fig. 2(c).

For the absorption signal near the $4p$ state, the electron can be excited from the ground state to IR-dressed $4p$ state, whose energy is close to field-free 4p state, by simultaneously absorbing one XUV photon and one 800 nm IR photon (XUV + IR process). In this case, we can replace ${\omega }_{\text{XUV}}$ in the above analysis with the required photon energy ${\omega }_{\text{XUV}}\approx E_{4p}-E_{1s}-{\omega }_{\text{IR}}=22.18\text{eV}$, whose position corresponds to the downward shift of the black-dashed horizontal line shown in Fig. 6(b), inducing the increase of the interval of the adjacent intersection. As a consequence, the offset $\Delta (\xi )$ becomes larger than the one in the $\text{2}p$ state case.

For the absorption signal near the continuum state $E_{\text{c}}=25.55\text{eV}$, the similar XUV + IR process requires the XUV photon energy ${\omega }_{\text{XUV}}\approx E_{\text{c}}-E_{1s}-{\omega }_{\text{IR}}=24\text{eV}$, whose position corresponds to the upward shift of the black-dashed horizontal line to the S point shown in Fig. 6(b). In this case, the offset $\Delta (\xi )$ is close to zero.

4. Conclusions

In conclusion, the ATAS in the IR laser-dressed helium atom is theoretically investigated by calculating single-atom frequency-dependent response function. The main interest is focused on the chirp effect of the attosecond pulse on the generated ATAS. By numerically solving both the fully 3D-TDSE and the three-level model, we demonstrate that although the structure of the absorption spectra is stable against varying chirp rate of the attosecond pulse, the attosecond chirp can induce a temporal shift of the absorption spectrum. We quantitatively identify that the chirp-induced shift distance relative to the chirp-free case is linearly dependent on the attosecond chirp rate and closely related with the IR pulse wavelength. Moreover, it is found that different absorption lines exhibit distinct dependencies on the attosecond chirp. In order to understand these features, we develop an analytical method based on the combination of the time-frequency analysis and the two-photon absorption picture and successfully reproduce the simulation result. Our result reveals that the attosecond chirp effect on ATAS originates from the indirect two-photon absorption processes through a set of the virtual states. This kind of the chirp effect might suggest a route towards the all-optical measurement of the chirp of the attosecond pulse.